Permutations Definition

A permutation is an arrangement of objects where the order matters. Two arrangements that use the same objects but in a different order count as different permutations. This lesson builds the counting principle that turns "how many orders are possible" into a short multiplication.

What you will learn

• Define a permutation as an ordered arrangement
• Distinguish order-matters from order-doesn't-matter counting
• Count arrangements by direct listing for small cases

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

What is a permutation?

A permutation is an ordered arrangement of some or all of a set of objects. If three friends line up for a photo, "Ana, Ben, Cy" and "Cy, Ana, Ben" use the same three people but count as two different permutations, because the positions they occupy are different. This is the key test for whether a problem is asking about permutations: swapping two objects while keeping the rest fixed produces a new, distinct outcome.

Contrast this with a situation where order does not matter — choosing 3 people out of 5 to form a committee, where the committee itself is the same regardless of the order the members were chosen in. That kind of counting is a combination, covered in a later lesson. For now, the signal to look for is language like "arrange," "line up," "order," "assign to positions," or "form a code," all of which mean that swapping two objects changes the outcome.

Counting without listing: the multiplication principle

Listing every arrangement works for very small cases, but it becomes impractical quickly — there are already 24 ways to arrange 4 distinct letters. The multiplication principle avoids listing: fill the positions one at a time, and at each position count how many objects remain available, then multiply those counts together.

For example, to arrange 4 distinct books on a shelf, the first position can hold any of the 4 books, the second position any of the remaining 3, the third any of the remaining 2, and the last position is forced to hold whichever single book is left. The total number of arrangements is $4\times3\times2\times1=24$. The same reasoning applies whether the objects are people in a line, digits in a code, or letters in a word: identify how many choices remain at each step, in order, and multiply.

Full arrangements versus partial arrangements

Sometimes every object gets placed (arranging all 4 books), and sometimes only some of the objects are placed into positions while the rest are left unused (choosing 2 digits out of 5 to form a 2-digit code). Both are permutation problems, and both use the same multiplication principle — the only difference is how many positions you are filling before you stop multiplying.

A useful habit is to draw one blank for each position you need to fill, then write above each blank how many choices remain at that point. This makes it easy to see whether the count of blanks matches the number of positions the problem actually describes, which is a common place for errors: filling too many or too few blanks changes the answer completely.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Treating an arrangement problem as a combination problem by forgetting that different orders count separately.
• Filling more or fewer blanks than the number of positions the problem actually asks for.
• Forgetting that once an object is used in one position, it is no longer available for the next position.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• A permutation is an ordered arrangement: swapping two objects produces a different outcome.
• Count arrangements by filling positions one at a time and multiplying the number of remaining choices at each step.
• Only fill as many positions as the problem actually asks for, even if more objects are available.